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Natural Science Forum / Physics / Relativity / November 2004Trying to understand a "paradox"
There's a chance this one is something already "out there", but I haven't seen it before. A friend gave me this "paradox" based on the train-and-embankment scenario, and although I can see where the problem lies, I can't see the actual problem... The train in this case mounts a pointer that moves vertically at a fixed rate based on it's own clock. On the embankment is a similar device geared to it's own clock. Each one includes a recording device that captures the position of the other's device and it's own when they pass. To make this "non-physical", we use lasers and photographic film, such that when the train passes the device on the embankment the train's laser records a streak on the embankment's film, and vice versa. There is negligable space between the two in either direction. At low speeds the train passes the embankment and the two have clocks that are running "in sync". Therefore they both end up at the same "height" when the pass, and comparing the film shows nothing interesting. Things are more interesting in the relativistic case. From the embankment's point of view the train's clock is running slow. Thus when they pass it would expect the train's laser to paint a line below it's own. Nothing too surprising there. From the train's perspective the embankment's clock is running slow, and therefore would expect it's line to be low. So if they later compare, it would seem that the beams are bent, the train records the embankment's line "down low" while the embankment itself records it "up high", and vice versa. I believe the problem revolves around the statement "when they cross". I've been trained to be leary of any statment containing an instant in time in relativity. Nevertheless I can't spot the EXACT problem in this setup -- or even if there is one. Maury Dear Maury Markowitz: ... > I believe the problem revolves around the statement "when they cross". > I've been trained to be leary of any statment containing an instant in > time in relativity. Nevertheless I can't spot the EXACT problem in this > setup -- or even if there is one. Actually the problem lies more in the unstated "when they start", because some form of simultaneity is required. David A. Smith > Dear Maury Markowitz: > [quoted text clipped - 6 lines] > Actually the problem lies more in the unstated "when they start", because > some form of simultaneity is required. I agree. Certainly there can be no dispute about the respective clock indications at the same point in time and space. In fact that would be the perfect point to synchronize the clocks! Harald >>Actually the problem lies more in the unstated "when they start", because >>some form of simultaneity is required. > > I agree. Certainly there can be no dispute about the respective clock > indications at the same point in time and space. In fact that would be the > perfect point to synchronize the clocks! Well OK, it appears I didn' think this through enough. So given this, what will be the outcome? From my "thought diagram" the train sees a short track with the embankment detector fairly close to it, which it will cross "soon". Therefore it's own projector will have only moved a small amount, and it expects the same for the one on the embankment. From the embankment's point of view the train has a "long" way to go, and will cross after both pointers have moved considerable longer distances. So one expects both lines to be near the bottom, the other expects them near the top. Furthermore, due to symmetry, they both expect the other line to be below their own (thus the "crossing" that is the "problem"). So without synced clocks I assume one of these will simply be running much earlier or later than the other. That can certainly remove the confusion between the "high" and "low", but it doesn't seem, at least off the top of my head, to solve the "crossing" problem. Further, it IS possible to sync clocks even in timelike separated frames, so again I don't see this as being an obvious solution to the problem. Maury Dear Maury Markowitz: ... > Furthermore, due to symmetry, they both expect the other line to be below > their own (thus the "crossing" that is the "problem"). [quoted text clipped - 7 lines] > frames, so again I don't see this as being an obvious solution to the > problem. Since you are sure they cross, then you need to work out why you feel that way. Keep in mind that in each frame's own film, its laser makes a vertical line. *Any* (co-planar) non-vertical line will eventually cross a vertical line... given enough film. David A. Smith > Since you are sure they cross, then you need to work out why you feel that > way. I am not sure they cross, and that's what I'm trying to figure out. At first glance, this does appear to be a problem. > Keep in mind that in each frame's own film, its laser makes a > vertical line. But that seems easy to avoid. Since the setup places the two detectors at epsilon from each other the signal time is also effectively zero -- a switch on the track would seem to be enough to change this. The key really does seem to be the nature of the "start time". However I still don't yet fully understand the problem here. So let's carry this discussion on... 1) Before starting, I sync the clocks so they both read midnight. I then travel, slowly, to the starting positions. 2) At noon on the train's clock I start rolling -- is it OK to assume that if the separation was at low speed, that the embankment's clock reads noon, or close enough to not be a real concern? 3) The detectors cross some time later, say a few seconds. T) I expect the train's clock to be something like 12:00:05, and the embankment's 12:00:10 -- using the assumption that the distance is covered much shorter as seen from the train's perspective (which may not be valid). Is the "experiment" outlined here OK? If the setup is OK, then I see the "crossing problem". However everyone here has focused on the syncing of the clocks, which leads me to believe that I don't fully understand the issue here. Is there a similar "syncing paradox" that is well-discussed? Maybe if I look at that I can see the problem here. Maury SNIP | The key really does seem to be the nature of the "start time". | However I still don't yet fully understand the problem here. So let's | carry this discussion on... | | 1) Before starting, I sync the clocks so they both read midnight. I then | travel, slowly, to the starting positions. OK, they are all in sync | 2) At noon on the train's clock I start rolling -- is it OK to assume | that if the separation was at low speed, that the embankment's clock | reads noon, or close enough to not be a real concern? Right | 3) The detectors cross some time later, say a few seconds. | | T) I expect the train's clock to be something like 12:00:05, and the | embankment's 12:00:10 -- using the assumption that the distance is | covered much shorter as seen from the train's perspective (which may not | be valid). Yes. Same from the embankment: the train's clock seems to run slow. In a nutshell, if I understand the paradox correctly: It's a bit paradoxical that the embankment's clock will turn out to be ahead of their own clock for observers in the train, despite original calibration and mutual time dilation. As interpreted from the embankment, what happens is that due to the train moving towards the embankment clock's light rays, the train occupants will estimate the embankment clock to be ahead of the train clock, right from the beginning (that is, IF the train occupants assume light speed to be homogeneously c relative to the train). Using your values, for a train instantly at full speed the embankment clock will seem to be 7.5 s ahead just after the start. Due to it seemingly tick twice as slow, it will indicate 12:00:10 when passing. | Is the "experiment" outlined here OK? If the setup is OK, then I see | the "crossing problem". However everyone here has focused on the syncing [quoted text clipped - 3 lines] | Is there a similar "syncing paradox" that is well-discussed? Maybe if | I look at that I can see the problem here. I hope to have sketched both the most likely paradox and its solution. Harald > As interpreted from the embankment, what happens is that due to the train > moving towards the embankment clock's light rays, the train occupants will > estimate the embankment clock to be ahead of the train clock, right from the > beginning Ok, this is the part I wasn't getting. In retrospect it seems obvious after reading some of the explainations of the twin paradox. As I understand it now, there are two cases: 1) you were always moving, and can't compare clocks 2) you started moving, at which point you go into a different reference frame, and see other clocks advance "all of a sudden". In this case we're considering (2), so as soon as the train starts rolling, it will see the other clock suddenly fast-forward. Thus when it arrives, there's no surprise that it reads further into the future. Is that basically "it"? If so, one more wrinkle if I may... Re-run the experiment as before. However this time sync three clocks on the embankment. Leave on where it was on the embankment, place one on the train (so far no changes), but place the third 1/2 way along the track. Start as before, but this time when the train reaches the clock "in the middle", sync to it. I'm sure the solution is similar, but i'd like to hear it anyway just so I'm sure I'm really understanding this and not just glancing on the surface. Maury > > As interpreted from the embankment, what happens is that due to the train > > moving towards the embankment clock's light rays, the train occupants will [quoted text clipped - 25 lines] > I'm sure the solution is similar, but i'd like to hear it anyway just so > I'm sure I'm really understanding this and not just glancing on the surface. Yes it makes no difference for the time sync procedure is linear, it just changes the zero point. And sorry I'm too busy to work it out for you, you can do it yourself for sure! Harald > >>Actually the problem lies more in the unstated "when they start", because > >>some form of simultaneity is required. [quoted text clipped - 24 lines] > frames, so again I don't see this as being an obvious solution to the > problem. Without synched clocks it depends on the history how the clocks are out of synch. It is possible to synch the clocks such that they will indicate the same when they pass each other. In that case we have again the symmetrical situation. Otherwise it's asymmetric. In fact, as in that example only one meeting point is defined, either you calibrate the clocks to be in synch when they meet there and the situation will look fully symmetric, or you don't and it won't be so - as you like! Harald > Actually the problem lies more in the unstated "when they start", because > some form of simultaneity is required. Perhaps, and I did consider that. However it's not clear how this changes the outcome -- which is that the two beams have "crossed". Maury > There's a chance this one is something already "out there", but I > haven't seen it before. A friend gave me this "paradox" based on the [quoted text clipped - 14 lines] > "height" when the pass, and comparing the film shows nothing > interesting. You need to consider how the two clocks were synchronised to start with if you want to compare their readings later. Martin Hogbin Martin Hogbin > There's a chance this one is something already "out there", but I > haven't seen it before. A friend gave me this "paradox" based on the [quoted text clipped - 32 lines] > > Maury They'll certainly agree on the positions of the pointers when they meet, but they won't agree at any other time (i.e. according to SR it's not possible to have agreement except when they're at the same location). > There's a chance this one is something already "out there", but I > haven't seen it before. A friend gave me this "paradox" based on the [quoted text clipped - 32 lines] > > Maury They don't have to cross. Only one experiences time slowdown. The train is the one that accelerated. There is no paradox. You can'y have it both ways. Einstein was wrong here to think it is reciprical. Its not. It only goes to show Einstein did not entirely understand time. But what can you expect? Mitch Raemsch -- Light Falls -- |
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